Scalar Einstein Equations

The Einstein equations $G_{\mu\nu} = 8\pi G T_{\mu\nu}$ express the metric evolution in terms of the matter sources. With the form of the scalar metric and stress energy tensor given in Eqns. (36) and (39), the ``Poisson'' equations become  

$$\begin{array} {rcl} k^2 \Phi &=& 4\pi G a^2 \left[ (\rho_f \... ...0)}\right), \vphantom{\displaystyle{\dot a \over a}}\end{array}$$ (56)

where the corresponding matter evolution is given by covariant conservation of the stress energy tensor $T_{\mu\nu}$, 

$$\begin{array} {rcl}\displaystyle{}\dot \delta_f & = & -(1+w... ...] + w_f k(\delta p_f/p_f - {2 \over 3} \pi_f ) \, ,\end{array}$$

for the fluid part, where $w_f = p_f/\rho_f$. These equations express energy and momentum density conservation respectively. They remain true for each fluid individually in the absence of momentum exchange. Note that for the photons $\delta_\gamma = 4\Theta^{(0)}_0$, $v_\gamma^{(0)}= \Theta_1^{(0)}$and $\pi_\gamma^{(0)}= {12 \over 5}\Theta_2^{(0)}$. Massless neutrinos obey Eqn. (60) without the Thomson coupling term.

Momentum exchange between the baryons and photons due to Thomson scattering follows by noting that for a given velocity perturbation the momentum density ratio between the two fluids is  

$$R \equiv {\rho_B+p_B \over \rho_\gamma + p_\gamma} \approx {3\rho_B \over 4\rho_\gamma} \, .$$ (57)

A comparison with photon Euler equation (60) (with $\ell=1$, m=0) gives the baryon equations as 

$$\begin{array} {rcl}\displaystyle{}\dot \delta_B & = & - kv_B... ...{\dot\tau \over R} (\Theta_1^{(0)}- v_B^{(0)}) \, .\end{array}$$

For a seed source, the conservation equations become

$$\begin{array} {rcl}\displaystyle{}\dot \rho_s &=& -3{\dot ... ...ver a} v_s^{(0)}+ k(p_s -{2 \over 3}\pi_s^{(0)})\, ,\end{array}$$

since the metric fluctuations produce higher order terms.